In a cellular MIMO communication environment, data rate can be increased through beamforming between a transmitting end and a receiving end.
It is determined based on channel information whether to use beamforming or not. Basically, the receiving end quantizes channel information estimated from a reference signal to a codebook and feeds back the codebook to the transmitting end.
A brief description will be given of a spatial channel matrix (also referred to simply as a channel matrix) for use in generating a codebook. The spatial channel matrix or channel matrix may be expressed as
      H    ⁡          (              i        ,        k            )        =      [                                                      h                              1                ,                1                                      ⁡                          (                              i                ,                k                            )                                                                          h                              1                ,                2                                      ⁡                          (                              i                ,                k                            )                                                …                                                    h                              1                ,                Ni                                      ⁡                          (                              i                ,                k                            )                                                                                      h                              2                ,                1                                      ⁡                          (                              i                ,                k                            )                                                                          h                              2                ,                2                                      ⁡                          (                              i                ,                k                            )                                                …                                                    h                              2                ,                Ni                                      ⁡                          (                              i                ,                k                            )                                                            ⋮                          ⋮                          ⋱                          ⋮                                                                h                              Nr                ,                1                                      ⁡                          (                              i                ,                k                            )                                                                          h                              Nr                ,                2                                      ⁡                          (                              i                ,                k                            )                                                …                                                    h                              Nr                ,                Ni                                      ⁡                          (                              i                ,                k                            )                                            ]  where H(i,k) denotes the spatial channel matrix, Nr denotes the number of Reception (Rx) antennas, Nt denotes the number of Transmission (Tx) antennas, r denotes the index of an Rx antenna, t denotes the index of a Tx antenna, i denotes the index of an Orthogonal Frequency Division Multiplexing (OFDM) or Single Carrier-Frequency Division Multiple Access (SC-FDMA) symbol, and k denotes the index of a subcarrier. Thus hr,t(i,k) is an element of the channel matrix H(i,k), representing the channel state of a tth Tx antenna and an rth Rx antenna on a kth subcarrier and an ith symbol.
A spatial channel covariance matrix R that is applicable to the present invention is expressed as R=E[Hr,tHr,tH] where H denotes the spatial channel matrix and E[ ] denotes a mean.
Singular Value Decomposition (SVD) is one of significant factorizations of a rectangular matrix, with many applications in signal processing and statistics. SVD is a generalization of the spectral theorem of matrices to arbitrary rectangular matrices. Spectral theorem says that an orthogonal square matrix can be unitarily diagonalized using a base of eigenvalues. Let the channel matrix H be an m×m matrix having real or complex entries. Then the channel matrix H may be expressed as the product of the following three matrices.Hm×m=Um×mΣm×nVn×nH where U and V are unitary matrices and Σ is an m×n diagonal matrix with non-negative singular values. For the singular values, Σ=diag(σ1 . . . σr),σt=√{square root over (λ1)}. The directions of the channels and strengths allocated to the channel directions are known from the SVD of the channels. The channel directions are represented as the left singular matrix U and the right singular matrix V. Among r independent channels created by MIMO, the direction of an ith channel is expressed as ith column vectors of the singular matrices U and V and the channel strength of the ith channel is expressed as σi2. Because each of the singular matrices U and V is composed of mutually orthogonal column vectors, the ith channel can be transmitted without interference with a jth channel. The direction of a dominant channel having a large σi2 value exhibits a relatively small variance over a long time or across a wide frequency band, whereas the direction of a channel having a small σi2 value exhibits a large variance.
This factorization into the product of three matrices is called SVD. The SVD is very general in the sense that it can be applied to any matrices whereas EigenValue Decomposition (EVD) can be applied only to orthogonal square matrices. Nevertheless, the two decompositions are related.
If the channel matrix H is a positive, definite Hermitian matrix, all eigenvalues of the channel matrix H are non-negative real numbers. The singular values and singular vectors of the channel matrix H are its eigenvalues and eigenvectors.
The EVD may be expressed asHHH=(UΣVH)(UΣVH)H=UΣΣTUH HHH=(UΣVH)H(UΣVH)H=VΣTΣVwhere the eigenvalues may be λ1 . . . λr. Information about the singular matrix U representing channel directions is known from the SVD of HHH and information about the singular matrix V representing channel directions is known from the SVD of HHH. In general, Multi-User MIMO (MU-MIMO) adopts beamforming at a transmitting end and a receiving end to achieve high data rates. If reception beams and transmission beams are represented as matrices T and W respectively, channels to which beamforming is applied are expressed as THW=TU(E)VHW. Accordingly, it is preferable to generate reception beams based on the singular matrix U and to generate transmission beams based on the singular matrix V.